where the product is taken over prime numbers p, and P(p, s) is the sum
In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that a(n) be multiplicative: this says exactly that a(n) is the product of the a(pk) whenever n factors as the product of the powers pk of distinct primes p.
In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region
that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.
In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here. The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GLm.
Examples
The following examples will use the notation for the set of all primes, that is:
The Euler product attached to the Riemann zeta functionζ(s), also using the sum of the geometric series, is
Using their reciprocals, two Euler products for the Möbius functionμ(n) are
and
Taking the ratio of these two gives
Since for even values of s the Riemann zeta function ζ(s) has an analytic expression in terms of a rational multiple of πs, then for even exponents, this infinite product evaluates to a rational number. For example, since ζ(2) = π2/6, ζ(4) = π4/90, and ζ(8) = π8/9450, then
and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to
where ω(n) counts the number of distinct prime factors of n, and 2ω(n) is the number of square-free divisors.
If χ(n) is a Dirichlet character of conductor N, so that χ is totally multiplicative and χ(n) only depends on n mod N, and χ(n) = 0 if n is not coprime to N, then
Here it is convenient to omit the primes p dividing the conductor N from the product. In his notebooks, Ramanujan generalized the Euler product for the zeta function as
for s > 1 where Lis(x) is the polylogarithm. For x = 1 the product above is just 1/ζ(s).
Notable constants
Many well known constants have Euler product expansions.
can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios (fractions where numerator and denominator differ by 1):
where each numerator is a prime number and each denominator is the nearest multiple of 4.[1]
G. Polya, Induction and Analogy in Mathematics Volume 1 Princeton University Press (1954) L.C. Card 53-6388 (A very accessible English translation of Euler's memoir regarding this "Most Extraordinary Law of the Numbers" appears starting on page 91)
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN978-0-387-90163-3, MR0434929, Zbl0335.10001(Provides an introductory discussion of the Euler product in the context of classical number theory.)